2 Answers
2

A hakira seeks to suggest two subtly-different explanations for something. If explanation A is correct, then X should apply in some other case; if explanation B is correct, then Y should apply instead.

It helps to think of different ways to view the subject. For instance:

First-come, first-served is how a Jewish burial society should bury its dead. Is this the general principle of "treat mitzvas first-come, first-served", or a special additional rule to honor the deceased? If the former, then a burial society that did otherwise does not owe the family an apology.

If my cow gored people three times, I now become especially responsible for its actions. Did the goring episodes prove that my cow was dangerous all along, or affect its temperament to turn it into a dangerous cow? The Mishna records a debate on what goring patterns count (does one brief episode of triple-goring count?), which may hinge on this distinction.

Can hiddur mitzva (doing the mitzva in a bonus, better way) be done in addition to the basic mitzva, or must it be part and parcel of it? If the latter, you can't do the mitzva then go back later and add the hiddur.

Literally it means "investigation" or something similar (and is used in other contexts, like that of evidence in court), but it's used to mean "hair-splitting".

Yes, really.

Well, almost really. A chakira is an investigation as to the exact nature of something, and is usually stated as a binary choice: is X an A or is X a B? (X can be an object, a state, an action, etc.) A practical difference is generally sought so that the distinction is clear and known to be real, and, where possible, proof is brought as to whether X is in fact an A or a B.